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Self-consistent Maxwell-Bloch model of quantum-dot photonic-crystal-cavity lasers

Cartar, William; Mørk, Jesper; Hughes, Stephen

Published in:Physical Review A (Atomic, Molecular and Optical Physics)

Link to article, DOI:10.1103/PhysRevA.96.023859

Publication date:2017

Document VersionPublisher's PDF, also known as Version of record

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Citation (APA):Cartar, W., Mørk, J., & Hughes, S. (2017). Self-consistent Maxwell-Bloch model of quantum-dot photonic-crystal-cavity lasers. Physical Review A (Atomic, Molecular and Optical Physics), 96(2), [023859].https://doi.org/10.1103/PhysRevA.96.023859

PHYSICAL REVIEW A 96, 023859 (2017)

Self-consistent Maxwell-Bloch model of quantum-dot photonic-crystal-cavity lasers

William Cartar,1 Jesper Mørk,2 and Stephen Hughes1

1Department of Physics, Engineering Physics and Astronomy, Queen’s University, Kingston, Ontario, Canada K7L 3N62DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, Building 345W, 2800 Kgs. Lyngby, Denmark

(Received 16 February 2017; published 28 August 2017)

We present a powerful computational approach to simulate the threshold behavior of photonic-crystalquantum-dot (QD) lasers. Using a finite-difference time-domain (FDTD) technique, Maxwell-Bloch equationsrepresenting a system of thousands of statistically independent and randomly positioned two-level emitters aresolved numerically. Phenomenological pure dephasing and incoherent pumping is added to the optical Blochequations to allow for a dynamical lasing regime, but the cavity-mediated radiative dynamics and gain coupling ofeach QD dipole (artificial atom) is contained self-consistently within the model. These Maxwell-Bloch equationsare implemented by using Lumerical’s flexible material plug-in tool, which allows a user to define additionalequations of motion for the nonlinear polarization. We implement the gain ensemble within triangular-latticephotonic-crystal cavities of various length N (where N refers to the number of missing holes), and investigatethe cavity mode characteristics and the threshold regime as a function of cavity length. We develop effectivetwo-dimensional model simulations which are derived after studying the full three-dimensional passive materialstructures by matching the cavity quality factors and resonance properties. We also demonstrate how to obtainthe correct point-dipole radiative decay rate from Fermi’s golden rule, which is captured naturally by the FDTDmethod. Our numerical simulations predict that the pump threshold plateaus around cavity lengths greater thanN = 9, which we identify as a consequence of the complex spatial dynamics and gain coupling from theinhomogeneous QD ensemble. This behavior is not expected from simple rate-equation analysis commonlyadopted in the literature, but is in qualitative agreement with recent experiments. Single-mode to multimodelasing is also observed, depending on the spectral peak frequency of the QD ensemble. Using a statistical modalanalysis of the average decay rates, we also show how the average radiative decay rate decreases as a functionof cavity size. In addition, we investigate the role of structural disorder on both the passive cavity and activelasers, where the latter show a general increase in the pump threshold for cavity lengths greater than N = 7, anda reduction in the nominal cavity mode volume for increasing amounts of disorder.

DOI: 10.1103/PhysRevA.96.023859

I. INTRODUCTION

Microcavity photonic crystals (PCs) are a natural advance-ment to mirror-based feedback systems as many technologiesshrink to the nanoscale. The vertical cavity surface emittinglaser (VCSEL) was successfully demonstrated in 1989 [1], andhas found applications in telecommunication systems, opticalinterconnects, spectroscopic sensing, and optical image pro-cessing. In 1994, Dowling et al. proposed a one-dimensional(1D) PC operating near the photonic band edge [2], makinguse of slow-light modes to increase the power emitted by suchlasers, which has been one of the limitations of microcavitylasers [3]. Experimentally, slow-light band edge lasers havenow been demonstrated in both two-dimensional (2D) [4–12] and three-dimensional (3D) [13–15] architectures, whileover the past decade, significant progress has been made inthe optimization of these lasers [15–18], allowing for theinvestigation of new operation regimes such as single emitterlasing [19], ultrahigh speed modulation [20], and self-pulsing[21]. To directly model the optical properties of open-systemmicrocavity structures, finite-difference time-domain (FDTD)techniques are often employed since such open cavitiessupport quasinormal modes (QNMs) that have a finite lifetimedue their coupling to a continuum of modes with outgoingboundary conditions [22]. For example, quasinormal modesare obtained from the mode solutions to the Helmholz equationwith open boundary conditions [23], resulting in a complex

eigenfrequency for each cavity mode. To numerically model again medium within the cavity, various techniques have beenimplemented ranging from the simple inclusion of a negativeimaginary component in the refractive index [24] to includingrate equations embedded in the FDTD algorithm [3,25,26],or with the finite element method [27]. It is also commonto adopt simple rate equations for the population density ofcarriers and photon flux [28,29], which can quickly connectto experimental data. It is, however, still a major challenge tomodel arbitrarily shaped gain materials coupled to arbitrarilyshaped cavity structures, which is desired for many quantum-dot (QD) microcavity structures, especially as the modalproperties of the laser cavity change drastically as a functionof position and size (which results in spatially dependentradiative coupling and gain dynamics). Semiconductor QDsare now increasingly used as the underlying gain materialin microcavity lasers, due to their superior room-temperatureoperation [30], tunability [31], unique atomlike density ofstates and carrier dynamics [32], and excellent temporal andspatial stability [30,33]. To develop a theoretical model ofthe light-matter interactions, one approach is to model theircollective gain more appropriately as an ensemble of effectivetwo-level atoms (TLAs) [34–36].

The simplest implementation of a TLA coupled to electro-magnetic fields is achieved with the optical Bloch equations(OBEs), which adds appropriate linear and nonlinear inter-actions between the dipole-induced polarization and electric

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WILLIAM CARTAR, JESPER MØRK, AND STEPHEN HUGHES PHYSICAL REVIEW A 96, 023859 (2017)

field, giving rise to the Maxwell-Bloch (MB) equations whencombined with the Maxwell equations [37]. Using an effectiveTLA, positive population inversion can be achieved, e.g., by aphenomenological incoherent pump rate that mimics ultrafastrelaxation rates from higher-lying levels to the lasing excitonstate, thus eliminating the need for additional energy levels.This simple model of an artificial atom, implemented withMaxwell’s equations, has the distinct benefit of allowing oneto study general light-matter interactions without using eitherof the rotating-wave approximation or the slowly-varying-envelope approximation, which have already led to a widerange of new effects such as the dynamic nonlinear skin effect[38] and carrier-wave Rabi flopping [39,40], even with simple1D equations of motion. Moreover, when studying quantuminformation systems that are dominated by radiative decay,it is critical to preserve the coherent radiative contributionsthat a MB analysis provides [41], without recourse to addingin phenomenological damping constants. Otherwise, if non-radiative processes dominate, a more straightforward MBformalism may be used, in which phenomenological dampingterms like pure dephasing are implemented [42,43]. DirectMB simulations have been successfully used for a numberof years. For example, Anderson and Cao used a 1D FDTDscheme to simulate stochastic noise in macroscopic atomicsystems [44]; Andreasen et al. carried out FDTD simulationof thermal noise in open cavities [45]; while Sukharev studiedthe interaction of chirped femtosecond laser pulses withhybrid materials comprised of plasmon grating structures andresonant molecules [46].

In many nanophotonic cavity structures, it is criticalto go beyond the 1D models and the commonly adoptedsimple rate equations; e.g., Sukharev and Nitzan have studiedatomic samples interacting with materials using a 2D MBmodel [47]; Pusch et al. studied amplification and noise ingain-enhanced plasmonic materials using a 3D model [48];Lopata and Neuhauser [49] studied the effect of nonlinearexcitations of a dipolar molecule on plasmon transfer acrossa pair of spherical gold nanoparticles using a split-fieldFDTD-Schrödinger approach; Gray and Kupka [50] carriedout FDTD studies of a variety of silver cylinder arrays withnanometer-scale diameters (nanowires) interacting with light;and Dridi and Schatz [51] introduced a model for describingplasmon-enhanced lasers that combines rate equations withFDTD for describing plasmon-enhanced lasers. Many of theradiative decay processes are also affected by unavoidablefabrication disorder, and even minute (nm-scale) levels ofdisorder can play an important role in understanding the richphysics of slow-light systems [52–58]. Deliberate disorder canbe added to a system to gain access to novel effects such asreduced laser thresholds in random laser systems [24], broughton by Anderson localization [59]. Indeed, there is continuedinterest in understanding Anderson localization modes andrandom lasing [60–62], which has recently been studied invarious disordered PC waveguide systems [56,58,63].

With the current trend of miniaturized semiconductor lasersystems, there is now a need for more sophisticated models ofPC lasers beyond the simple rate-equation picture [28,64],where the emitters’ coherence is assumed to be in steadystate or adiabatically eliminated, leading to coupled equationsbetween the available energy levels without any information

L5 cavity

L5

L7 cavity

M1

M1

FIG. 1. Top-down view region of an L5 (a) and L7 (b) PC slabcavity. (c) First fundamental mode (M1) profile |E(r)|, inside an L5cavity, computed by 3D FDTD simulations. The circle outlines whereetched holes exist within the homogeneous background material,which has an index of refraction of n = 3.17 in 3D simulations. Thetwo white lines in the x − z spatial profile represent the characteristiclength lz,V we use to relate 2D mode volumes to 3D calculations (seeSec. II). The slab has lattice pitch a = 438 nm, radius r = a/4, andheight h = 250 nm. The mode profile measured at the center of the3D slabs (in the z direction) is very similar to an effective 2D modeprofile with the same properties, except n = 2.54 (see text).

regarding the system coherence. In many cases, this mayfail to describe emerging experiments. For example, a recentinvestigation of lasing threshold as a function of band-edgeproximity, performed by increasing a triangular lattice PCcavity length, found counterintuitive results [13]: rather than adecreasing gain threshold for increasing cavity length, whichis predicted by simple laser theory, there existed a thresholdminimum around the L8-L9 cavity length scale, where LN

denotes a cavity of length N (missing holes in the lattice),and cavity lengths ranging from L3–L20 were created. Twoexample schematics are shown in in Figs. 1(a) and 1(b).These cavities have fundamental cavity modes (M1, M2, . . .)confined within the cavity region, as shown in Fig. 1(c) foran L5 cavity. In the theoretical analysis of a 2D square-basedPC microcavity laser, a similar trend is observed, but only forsystems with low optical density of states (DOS) [3]. Since theDOS and LDOS (local DOS) of band-edge cavities are so high,the results of Ref. [13] were partly explained by a heuristicmodel of disorder-induced backscattering and outscatteringof the Bloch mode into modes above the light line (whichis known to occur in longer-length PC waveguides). In sucha model, disorder shifts some of the lasing mode near theband edge into the regime where it is no longer confined tothe cavity structure, and this shift is felt more strongly by

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longer cavity modes (slower light is more sensitive to disorder),which exist deeper in the slow-light regime, thus creating anoptimal pump threshold by minimizing both reflection lossesand backscattering losses.

In this paper, we present a systematic numerical study ofa QD ensemble in triangular-lattice PC cavities, and explorethe lasing threshold behavior as a function of PC length. Weclosely follow the designs and recent experiments of Xue et al.[13], and also partly explore the role of structural disorder onthe lasing threshold in PC cavities, which were credited to belikely responsible for the unusual gain threshold dependenceon cavity length. Using Lumerical’s FDTD material plug-intool, which allows a user to include unique polarizations insimulation objects [65], we investigate the gain and lasingbehavior of effective 2D cavity laser structures, modeledafter full 3D passive simulations. Although we use LumericalFDTD, the general technique can naturally be adopted with anygeneral FDTD (or time-dependent Maxwell) solver. The user-controlled plug-in tool returns MB dynamics by solving theOBEs (for each QD), and includes radiative decay, local gain,and inter-QD coupling that is fully captured self-consistentlyby the FDTD method, as well as pure dephasing includedas a phenomenological decay rate, and an incoherent pumpterm which effectively models a three-level gain system foreach QD. The OBE plug-in has the distinct advantage of beingcompletely general, solving lasing dynamics and gain couplingwith zero a priori knowledge (other than inherent propertiesof the QDs), when compared to traditional rate equations[28,29], and they easily capture the statistical behavior of aQD ensemble as well.

The remainder of our paper is organized as follows: InSec. II, we introduce the cavity parameters and model thebasic cavity properties of the 2D simulations after obtainingthe results for passive 3D (slab) structures. In addition, weinvestigate the role of fabrication disorder of the PC lattice,and model 2D after 3D simulations once more. In Sec. III, weintroduce our model OBEs, and discuss their implementationwithin the plug-in tool as a source for nonlinear polarization. InSec. IV, we discuss the dipole moment used in our simulationsand explain how to obtain the correct 3D radiative decayof a point dipole in an effective 2D model. In Sec. V, wediscuss the implementation of QDs in the FDTD method, andmodel our radiative decay after 3D simulations. In Sec. VI,we outline and discuss the results of including an active QDensemble (including 14 000–24 000 randomly positioned QDswith random center frequencies) for various cavity lengths,extracting pump thresholds, and investigating different modelsof the plug in. We connect our results to the recent experimentsof Xue et al. [13] and standard rate equations, and provideinsights into the gain threshold dependence on cavity length.We summarize in Sec. VII. In the Appendix, we exemplifythe role of structural disorder on the gain threshold andlasing modes, and highlight a number of effects such as modelocalization for increasing disorder.

II. PASSIVE CAVITY SIMULATIONS: EFFECTIVE 2DSIMULATIONS AND ROLE OF FABRICATION DISORDER

Full 3D simulations of passive PC slab structures (i.e., withno gain material) form an appropriate starting point to model

N N

M1

FIG. 2. Passive cavity simulations for an LN cavity in a triangularPC cavity with lattice pitch a = 438 nm, radius r = a/4, heighth = 250 nm (in 3D), and refractive index n = 3.17 or 2.54 for 3Dand 2D simulations, respectively. (a) Comparison of the fundamentalmode’s peak frequency as a function of cavity length, for 2D and3D simulations, with experimental data points from 13 for reference.(b) Run-time requirements using 16 cores with 1024 Mb of memory,for a single passive 2D or 3D simulation, without QD meshrequirements, as a function of cavity length (see text).

planar PC slabs with no active QDs, as they produce the mainmode characteristics and allow for additional key parameterssuch as Q,Veff , and local density of states (LDOS). However,2D simulations take significantly less time to run, and can bemodeled after 3D simulations to capture the key propertiesof PC slab modes with similar peak properties, such as modefrequency [shown in Fig. 2(a)] and Q factors (studied later),at a fraction of the computational cost [see Fig. 2(b)]. Thisis important for developing effective 2D models for the fullMaxwell OBEs with gain materials and many thousands ofOBEs (i.e., one OBE set for each QD). It should also be notedthat Fig. 2(b) is obtained for passive structures only, and toaccurately include QD dipoles in our simulation, we requirea finer spatial mesh of more than twice what is typically usedin passive simulations. As such, the simulations performedwith a QD ensemble require significantly longer to run thanis represented in Fig. 2(b), so that the lasing dynamics mayeventually reach steady state (SS). For example, each of our 2DL15 lasing simulations (shown below) takes roughly 20 hoursto run, when 16 computational cores are used with 1024 Mb ofmemory each. This increased run time is roughly 400 times thepassive simulations shown in Fig. 2(b). As such, for this firststudy, we chose to develop an effective 2D FDTD method whenusing the OBEs, which makes it easier to carry out a systematicsweep of various system parameters such as cavity length andpump powers, especially important for high-Q cavity modeswhich take a long time to reach SS.

In order to introduce an accurate effective 2D cavitysimulation, similar to the PC cavity experiments of Xue et al.[13], we first capture the basic cavity physics using passive3D slab simulations. The cavities are assumed to be madeof InP, with a standard hexagonal lattice PC cavity, withoutany hole shifts or modifications to optimize the cavity Q.The lattice pitch is a = 438 nm, with hole radius r = a/4,slab height h = 250 nm, and refractive index n = 3.17 [13].The PC band gap (TE-like) is roughly 185–215 THz, and theQDs have parameters similar to InAs. Our simulations are run

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WILLIAM CARTAR, JESPER MØRK, AND STEPHEN HUGHES PHYSICAL REVIEW A 96, 023859 (2017)

N

ML

FIG. 3. (a) Impact of intrinsic levels of structural disorder (redcircles) on 3D FDTD PC cavity simulations as a function of cavitylength, averaged over 20 instances (red dotted-dashed line), comparedto ordered simulations (dashed black line). (b) Schematic top viewof the disordered cavity simulation with the disordered holes (redcircles) over-top the ideal air holes (black holes), and the backgroundslab (gray); this is a zoom-in region and in the simulations use a largerdomain seen in Fig. 5(b).

using Lumerical’s [65] FDTD software, with open boundaryconditions via perfectly matched layers (PMLs).

An example of the L5 cavity is displayed in Figs. 1(c)and 3(b). We simulate cavity lengths ranging from L5–L15skipping even cavity lengths, and measuring the resulting Q

factors for all fundamental modes observed in the simulations.The cavity modes are excited by a dipole source defined bya fixed carrier frequency with a temporal Gaussian envelope,located along the center axis of the cavity, shifted from thecentral y axis to avoid emitting at the antinode points ofeven cavity modes (i.e., M2, M4, etc.). To capture the modalproperties of each cavity, two simulations are run; the firstsimulation uses an electric-field time monitor to measure thefirst few dominant mode eigenfrequencies ωμ = ωμ − i�μ,where ωμ is the peak frequency, and �μ = ωμ

2Qμis defined by the

mode broadening and the cavity quality factor Q; the secondsimulation measures each mode’s spatial QNM profile f(r)using a discrete Fourier transform (DFT) monitor. Typically,three–five modes are measured in the frequency range ofinterest, depending on the cavity length, as longer cavities havestronger higher-order modes, and more frequencies within thesimulation bandwidth.

Using the cavity mode profiles and their correspondingeigenfrequencies, we can calculate the cavity Green function(GF) using a QNM expansion [66]

G(r,r′; ω) =∑

μ

ω2 fμ(r)fμ(r′)2ωμ(ωμ − ω)

, (1)

where μ uniquely identifies each mode, and the modes arenormalized by [67]

〈〈fμ|fμ〉〉 = limV →∞

∫V

εr (r)fμ(r) · fμ(r)dr

+ ic

2ωμ

∫∂V

√εr (r)fμ(r) · fμ(r)dr = 1, (2)

where εr (r) is the relative permittivity of the cavity, and ∂V

denotes the border of volume V , in the appropriate limit [68].With the normalized modes, we are able to calculate accurateeffective mode volumes [67], which are defined from

V −1eff = Re

{εr (rc)f2

c (rc)

〈〈fc|fc〉〉}, (3)

where rc is an antinode point of interest within the cavitystructure. Having the GF defined at all locations of r and r′allows us to, e.g., plot the LDOS at any location (i.e., withouthaving to do further dipole calculations), and normalizing bythe free-space GF defines the projected LDOS to have unitsequivalent to the Purcell factor (PF), defined as [69]

PF = 3

4π2

(λ

n

)3Q

Veff, (4)

where λ is the wavelength and the Q is for the resonant mode ofinterest. This expression characterizes the enhanced radiativedecay with respect to a homogeneous medium and assumes adipole in resonance with a single cavity mode and perfectlymatched to the field maximum and polarization. Furthermore,if the dipole is exposed to pure dephasing, the expressionassumes that the dephasing rate is much smaller than the cavitydecay rate (which we note is not the case for the simulationstudies later) [64].

Next, to identify the role of fabrication or structural disorderon the passive 3D structures, we model intrinsic fabricationdisorder by shifting the center of each hole by a random amount�r characterized by the standard deviation σDis of a randomnumber generator. The direction of each hole’s shift is alsorandomized, by defining a random number between [−π,π ],thus giving equal probability for a shift in any direction.Figure 3(b) depicts an exaggerated disorder instance for anL5 cavity. The required numerical size of the PC (to mimican infinite PC system) was determined by increasing thesimulation’s spatial size until the largest (dominant) Q-factorvalue converged, and the simulation size increased as the cavitylength increased to prevent spurious Q-factor measurements.The intrinsic disorder is set to be σDis = 0.005a, as determinedfrom related experimental far-field intensity spectra comparedto FDTD simulated spectra for varying amounts of disorder[70]. Figure 3(a) depicts the measured Q factors of 20 instancesof disorder at each cavity length, showing the impact ofdisorder is very minimal at smaller cavity lengths, and impactsthe L15 Q factor by only about 15%. These disorderedstatistics are consistent with the other findings of similar PCcavity investigations [71], for the same range of Q.

Given the measured eigenfrequencies and (effective) modevolumes for passive 3D simulations, effective 2D simulationsare subsequently optimized to closely match the Q factor,mode volume, and peak frequency trends of the full 3D sim-ulations. First, the 2D simulations use an effective refractiveindex of n = 2.54 to shift the peak frequencies to match the3D simulations, which use n = 3.17. Fitting this effectiverefractive index optimized both the location and separationbetween the first and second fundamental modes M1 andM2, respectively, so that any mode coupling affects wouldbe representative of their 3D counterparts. This is mainly whyFig. 2(a) is not simply an optimized overlap between 2D and

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NN

N

FIG. 4. (a)–(c) Q and Veff modeling of effective 2D systemcompared to full 3D simulations, without disorder. (d) Disorderstatistics of �Dis = � − �0, for the L15 cavities of varying amounts of2D disorder, referenced to the 3D intrinsic disorder of σDis = 0.005a.

3D simulation trends. In addition, to account for the leakagethat occurs in 3D simulations (in that case through verticaldecay), the size of the 2D PC in the x direction was set to allowcavity decay that was similar to the 3D leakage. This can beseen in Fig. 4(a), where the 2D and 3D Q factors are compared.In Fig. 4(b), we also compare 2D and 3D mode volumes,which requires a characteristic length scale lz,V to convert the2D mode area Aeff,2D into a volume Veff,2D = lz,V Aeff,2D. Thislength scale is chosen to be 205 nm [see Fig. 1(c) in Sec. I],which is slightly less than the height of the 3D PC cavities. Wechose this length scale as it gives a reasonable Q/Veff fit, asshown in Fig. 4(c), while also containing the majority of the3D mode volume (more than 85%). Given the complexity ofthe calculations that follow, this is an appropriate effective 2Dmodel to capture the 3D slab cavity effects.

To appropriately model fabrication disorder in the effective2D model, we compared the statistical average and varianceof the L15’s 3D simulation to varying amounts of structuraldisorder for the corresponding 2D simulation. Taking thenormal definition of our Q factor to be Q = ω

2�, we can

capture the disorder statistics using �, as ω is roughly constantwith increasing disorder. Defining � = �0 + �Dis, where �0 isthe ideal structure’s broadening [full width at half-maximum(FWHM)] and �Dis is additional broadening due to structuraldisorder, we plot 〈�Dis/�0〉 for the 3D data collected inFig. 3(a) and compare it to 100 instances of 2D simulationswith σDis = [0.0025,0.005,0.01,0.02] (400 simulations total),

as shown in Fig. 4(d). In our 2D simulations, intrinsic disorderis seen to be best modeled (namely, more similar to 3D) byσDis = 0.01a.

III. EFFECTIVE TWO-LEVEL ATOM MODELAND POLARIZATION PLUG-IN EQUATIONS

Here we describe a simple effective gain model for typicalexperimental QDs [72]. We assume QDs that can be describedas effective TLAs, where the physics of higher-order energylevels is effectively ignored (or adiabatically eliminated), andwe use an incoherent pump term P to create a positivepopulation inversion and thus gain [73]. While one couldimplement an optically driven multilevel system [48], in realitythe level structure of the QDs is quite rich and vary from dot todot, so it would not be very meaningful; in addition, as long asthe relaxation rates from the higher-lying levels are sufficientlyfast, an effective two-level model is expected to be accurateand sufficient for our current study.

To derive the OBEs, we use a master equation to solve forthe density matrix of each TLA, and treat the electromagneticfield classically. Starting with the system Hamiltonian of aTLA, with a dipole moment d defined by a ground state|0〉, and excited state |1〉, with energy difference ω0, andinteracting with an electromagnetic field E, we have the systemHamiltonian

H = hω0

2σz − h(t)(σ+ + σ−), (5)

where the Rabi frequency (t) = d(t)·E(t)h

describes the fieldinteraction with the dipole moment, σz is the z− componentPauli matrix, and σ+ and σ− are the raising and loweringoperators of our TLA, respectively. We stress that the electricfield E(t) is solved self-consistently by FDTD (including thedipole field), while d and ω0 are set by the material plug-inequations, which solve the OBEs derived from the masterequation of this Hamiltonian. One can think of this Rabi fieldas the local or self-consistent Rabi field. Since our QDs ofinterest are modeled at room temperature, the dominant sourceof damping is due to nonradiative processes, in particular puredephasing. The dissipative nature of our QDs environmentis included phenomenologically using Lindblad superoper-ators L, defined as L(O)ρ = OρO† − 1

2 (O†Oρ + ρO†O).Traditionally, when a TLA interacts with degrees of freedomsuch as photons, phonons, other collective modes, molec-ular vibrations, rotations, and translations, it experiences abroadening of its absorption linewidth directly proportionalto the total dephasing rate [74]. This broadening has twomain contributions in QDs: an inherent relaxation rate �R

determined by the TLA’s environment (e.g., for maximumcoupling �R ∝ Q/Veff) captured by the FDTD method atall positions, and a pure dephasing rate γ ′ which is relatedto the temperature, and to coupling to lattice vibrationsin the solid (phonons), and charge noise. We add puredephasing phenomenologically to our system Hamiltonian viathe Lindblad superoperator γ ′L(σ+σ−).

To achieve positive population inversion, we include theLindblad term PL(σ+), which is responsible for pumping theexcited QD lasing state. We neglect any influence from thepump field on additional dephasing [73] (since our largest P

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WILLIAM CARTAR, JESPER MØRK, AND STEPHEN HUGHES PHYSICAL REVIEW A 96, 023859 (2017)

is around 8 ns−1, while our γ ′ is 1.5 ps−1). In the usual way,we use the quantum Liouville equation ρ = −i

h[H,ρ] with the

Lindblad terms to derive equations of motion for the densitymatrix. The master equation is now given by

dρ

dt= −i

h[H,ρ] + PL(σ+)ρ + γ0L(σ−)ρ + γ ′L(σ+σ−)ρ,

(6)

where γ0 ≈ 0.1�hom [75] (�hom ≈ 0.4 ns−1 is the decay ratein a homogeneous medium, defined later) is the decay rateassociated with out-of-plane decay (i.e., via radiation modesabove the slab light line), ρ is a 2 × 2 matrix (for each QD) withdiagonal elements ρ11 and ρ00 associated with the probabilitiesof being in the excited and ground states, respectively, andoff-diagonal elements ρ01 = ρ∗

10 associated with the system’scoherence. We define inversion as ρ11 − ρ00 = 2ρ11 − 1 sinceρ11 + ρ00 = 1. Solving Eq. (6), we find equations of motionfor ˙ρ11 and ˙ρ01:

dρ11

dt= −2ρIm

01 + P (1 − ρ11) − γ0ρ11, (7)

dρRe01

dt= −ω0ρ

Im01 − P + γ ′

2ρRe

01 − γ0

2ρRe

01 , (8)

dρIm01

dt= ω0ρ

Re01 − P + γ ′

2ρIm

01 + (2ρ11 − 1) − γ0

2ρIm

01 , (9)

where we have separated ρ01 into its real Re[ρ01] = ρRe01 and

imaginary Im[ρ01] = ρIm01 parts, thus leaving three coupled

equations with only real parameters (per QD); this is done fornumerical convenience. Together, these equations define theoptical Bloch equations (OBEs) which describe the quantumnature of an effective TLA interacting with a completelygeneral classical electric field via FDTD. This formalismneglects a small frequency shift from the self-field of theemitter, which arises due to the numerically divergent in-phasecontribution at the location of the emitter [76], though this hasnegligible influence on our findings, especially with eventualrandom center frequencies for each QD emitter.

In deriving Eqs. (7)–(9), we have not made any ap-proximations other than those defined by the model itself(i.e., only two energy levels). This is unusual compared tostandard textbook derivations [77,78], which often invokea rotating-wave approximation; however, this is done as aresult of assuming some form for the electric field, whereaswe have not assumed any information regarding our quicklyvarying electric field. Instead, we leave our OBEs quitegeneral, such that the FDTD algorithm captures the light-matter physics experienced by our TLA in a self-consistentway, including radiative decay and dipole-dipole interactionsbetween different QDs. To include the OBE in the FDTDsimulations, we use Lumerical’s user-defined-material plug-intool [79], which allows for the creation of customized materialresponses, written in C++. The plug-in code is called at eachtime step n, and is used to update the electric field En bythe polarization density Pn output by the plug-in script. Ingeneral, this is written as UnEn + Pn

ε0= V n, where Un and

V n are inputs provided by Lumerical’s software, and En isupdated along the x, y, and z axes.

To determine the polarization density P output by ourOBEs, we use p = er(ρ01 + ρ∗

01) = 2er Re[ρ01], such that thepolarization density P = Np is given by

P = 2dN Re[ρ01], (10)

where er = d is the dipole moment associated with our atom,which assumes our dipole moment is equally strong in alldirections, while N is the relevant density of the QD beingpolarized, defined by

N = V −1QD = δ(z − z′)/(�x�y), (11)

where V −1QD is the inverse QD volume (i.e., VQD is the same for

each QD, and is defined by the Yee cell volume, or area in this2D case), and �x, �y are the Yee cell dimensions of the FDTDsimulation. We note that the assumption to have an equallystrong dipole moment in all directions is not representativeof natural QD ensembles, and future works could includestatistical variations of the dipole moments direction. However,since the optical modes are strongly polarized in a particulardirection, we do not expect that it would make much differenceto our simulations below. Finally, to implement the OBEsnumerically, we use the fourth-order Runge-Kutta method,thus storing only the previous time-step values ρn

11, ρIm,n01 , and

ρRe,n01 , when updating ρn+1

11 , ρIm,n+101 , and ρ

Re,n+101 , and P n using

Eq. (10).

IV. DIPOLE MOMENT MODEL AND RECOVERING THECORRECT RADIATIVE DECAY FROM A POINT DIPOLE

In the dipole approximation, where we assume thatinteracting electromagnetic fields have negligible variationover each quantum emitter, we define the oscillator strengthF of the InAs QD, as F(ω) = �rad,hom(ω)/�HO(ω) [80],where �rad,hom(ω) is the homogeneous medium’s radiativedecay rate, which includes information about the excitonicdegrees of freedom, and �HO(ω) is the radiative decay rateof a classical harmonic oscillator of elementary charge. Thisapproximation is valid because of the typical small size of theQDs, ∼15 nm in diameter (and a few nm in height) [72].Following Ref. [80], F(ω) is proportional to independentelectron and hole envelope function Fe(r0,r), and Fh(r0,r),respectively, in the strong confinement regime, given byF(ω) = Ep

hω|∫ d rFe(r0,r)Fh(r0,r)|2, where Ep is the Kane

energy of the QD material. The wave-function overlap IWF =|∫ dr Fe(r0) rFh(r0,r)|2 is a relatively constant function withrespect to QD size, and is mainly dependent on the emissionenergy of the QD [81]. A QD emitting at 190 THz (0.79 eV),which has an electron-hole overlap IWF > 0.8 [81]. In addition,the Kane energy of bulk InAs to be 21.11 eV [82]. Thus, weassume QDs with an oscillator strength of roughly 17.2 whenassuming a wave-function overlap of 80%.

The dipole strength d of our modeled QDs can be calculateddirectly by the oscillator strength, given by [83]

d2 = e2h

2mωF , (12)

where e is the elementary charge and m is the free-electronmass. Using our oscillator strength F = 17.2, we calculate adipole moment of 43.88 D (or 0.91 e-nm).

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L

FIG. 5. (a) Quantum-dot dipoles located at Ey along the Yeecell, randomly positioned with no nearest neighbors, with a densityof 540 μm−2. The grid dimensions are dx = 16.8462 nm anddy = 14.5892 nm, while the QD size is made smaller than this toemphasize its location at only a single grid point; however, its size iseffectively dx dy. (b) Quantum-dot gain setup: the orange border isPML boundary conditions, the gray bar on the left is the initial sourcepulse, which stimulates the optically pumped QD gain region in thecenter, the yellow rectangle denotes the index and DFT monitors,the yellow cross is a time monitor, and the orange line on the rightis a power line monitor that captures the average power out of thecavity, over the last 10 ps. The total number of QDs in this particularsimulation is 14 029 (for reference, L15 simulations have ≈24 000QDs).

V. QUANTUM-DOT MODELING IN FDTDAND SIMULATION SETUP

To model QDs in FDTD, the TLA plug-in is implementedat a single Yee cell dipole with an area (in 2D) equal to theYee cell. We choose to implement QDs at Ey field points,seen in Fig. 5(a), as our cavity modes primarily exist in thisdirection. To better model the QD dimensions, we reduce themesh steps �x,�y to 16.85 and 14.59 nm, respectively, whichis asymmetric to maintain uniform meshing across all etchedholes (in a triangular lattice). The QDs are then added to ourgain region randomly, with an area density NQD = 540 μm−2.The only restriction placed on the QD locations is that no twoQDs may be side by side. Each QD field location is given a

background index that matches the substrate material, which isa good approximation since the QD material has an index verynear the slab structure. To ensure that no QD is created withinthe etched holes of the PC, a mesh order is assigned to the QDsto be the second last material added to the system, with etchingas the final material, added over-top of all previous indices.

An example simulation setup is shown in Fig. 5(b), whichhas the following simulation features: a time monitor at itscenter, an index, and a DFT monitor around the cavity inyellow (inner rectangle) to capture the electric field profiles, again region made as small as possible (e.g., the smallest regionwith the same steady-state output as larger gain regions) to savemeshing memory overhead, an incident plane-wave pulse onthe left, a power line monitor on the right, and PML boundaryconditions all around. The plane wave is angled slightly toexcite both Ex and Ey field components, and the power linemonitor only captures the last 10 ps of power emitted by thecavity. This is consistent across all cavity lengths simulated,while the time each cavity simulation is run for is determinedby how long it takes to reach steady-state lasing.

Since real QD materials have a large fluctuation in theQD emission frequencies, each QD has its energy spacing ω0

randomly drawn from a normal distribution to better representslight variations in the QD size that occurs in practice.Thus, nonuniform emission lines lead to inhomogeneousbroadening, which can be modeled with parameters obtainedfrom experimental data. At room temperature, InAs QDs aredominated by pure dephasing γ ′, typically around 1.5 THz(or 6 meV) [33], and the overall inhomogeneously broadenedspectra are roughly 10 THz (or 40 meV), as shown by theQD ensemble electroluminescence in Ref. [72]. Thus, weassign each QD a resonant frequency that is selected randomlyfrom a Gaussian distribution modeled after experimentalphotoluminescence spectra. This model is shown in Fig. 6(a),where the Gaussian distributions variance is 6.6 THz, withmean ω0. In this way, both QD position and resonant frequencyare stochastically modeled.

To further connect our 2D simulations to the 3D dipoleinteractions, we first model radiative decay using Fermi’sgolden rule in 3D, which is is well known [84]:

�a(ra) = 2

hε0{da · Im[G(ra,ra; ω)] · da}, (13)

where G is then projected in the dominant field direction (inour PC cavities, that is the y direction), at the location ofthe dipole emitter. This definition of radiative decay assumeswe are in 3D space, and our GF has units of inverse volume(m−3), while the dipole moment has units Coulomb meter(Cm), and the overall decay rate has units of inverse seconds(1/s). Assuming radiative decay is calculated with the sameformula in 2D, it is then required to introduce an effectivelength lz as our GF loses a spatial dimension from 2D to 3D.That is, we define

�2D,eff = �3D, (14)

where �2D,eff is the radiative decay rate obtained in the 2Dsimulations. To achieve this, we use the radiative decay ratesof free space such that �2D,free space/lz = �3D,free space, which

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FIG. 6. (a) Modeled QD ensemble (dashed and dotted-dashedlines) for 10 (green chain) and 10 000 (magenta dashed line)QDs, for typical room-temperature inhomogeneous broadening. Theindividual QD line shape is given by the orange (light) line, withγ ′ = 1.5 THz. The y-shows a normalized photoluminescence (PL)for individual (solid line) and ensembled (dashed or dotted-dashedlines) QDs. The number of QDs in each simulation is on the orderof 10 000. (b) Fundamental mode comparison between full-dipoleexpansion (blue line) and QNM theory (orange dashed line) for low-Qcavity, with a reduced PC membrane. The projected LDOS has beennormalized to represent PF.

defines an effective length lz as

lz = �2D,free space

�3D,free space= G2D,free space

G3D,free space. (15)

The free-space Green function in 2D is derived asIm{Ghom

2D } = 18

ω2

c2 (for TE modes), while in 3D we have

Im{Ghom3D } = n

6πω3

c3 [85]. Using these definitions in Eq. (15),then

lz = 3π

4n

c

ω, (16)

and the effective radiative decay rate in our simulations is

�2D,eff(r) = 8nω

3πhε0c{d · Im[G2D(r,r; ω)] · d}. (17)

To verify that this model indeed obtains the correct radiativedecay from a dipole, we scale our plug-in density Neff bylz in Eq. (16), such that N−1

eff = �x�ylz, arriving at Neff =1.743 × 104 μm−3 for ω = 190 THz. We then model our plug-in material at a single Yee cell at the center of an L5 cavity, andemploy an initially excited state to measure the natural decayof the TLA, and turn off phenomenological decays γ ′, γ0, andthe incoherent pump P . Thus, there are no phenomenologicalterms in the OBE at all, either radiative or nonradiative. We setthe dipole moment magnitude to 43.88 D to mirror the valuesthat will be used in the ensemble simulations below. To initiallyensure we are not near the strong-coupling regime, we first setup our 2D cavity to have a low-Q factor, by shortening the PCmembrane on either side of the cavity edge. We give the TLAan initial polarization to simulate a dynamical decay from aradiation reaction (which is known to give the same decay asfrom vacuum fluctuations). From this simulation, we find verygood agreement between theory and simulation, where the GFof our cavity is calculated with Eq. (1), which is shown to

FIG. 7. (a) Population ρ11 dynamics within an L5 low Q-factor(≈400) cavity, solved by MBE without any decay terms, so allphenomenological decay parameters are set to zero and the radiativedecay is captured self-consistently through the local electric field usedin the OBE solver. The OBEs are given initial conditions ρ11(t = 0) =0.95, ρIm

01 (t = 0) = 0, and ρRe01 (t = 0) = √

1 − 0.952/2, such that theBloch vector magnitude is set to unity (e.g., σ1 = 2ρ11 − 1, σ2 =2ρRe

01 , and σ3 = 2ρIm01 are the usual Bloch vector components with

magnitude σ 21 + σ 2

2 + σ 23 ). The analytic solution is determined by

�2D,eff , calculated by the QNM GF seen in Fig. 6(b). (b) Comparisonof high- and low-Q cavities for the 2D L5 cavity structure, whichwere 47 000 and 430, respectively. All other parameters are the sameas in (a).

agree with numerically exact (i.e., full-dipole) simulations inFig. 6(b). Importantly, this accuracy will be maintained at anyspatial position within our simulation array of multiple dipoles.We also stress that the FDTD also captures QD-QD radiativeinteractions [86], which in a master-equation approach wouldbe computed from terms like

�ab(ra → rb) = 2

hε0{da · Im[G(ra,rb; ω)] · db)}, (18)

and a corresponding Lamb shift [87]. Clearly including suchterms in an ensemble of different QDs, in the many thousands(and even in the tens), would be intractable in a standardmaster equation approach. Dipole-dipole coupling has recentlybeen shown to have a strong impact on the subradiance andsuperradiance on steady-state QD laser systems [88], thusincreasing the importance to model it correctly in such systems.

To fit our simulations for the single dipole �Num with �2D,eff ,we match decay lifetimes as seen in Fig. 7(a). Note, the initialdecay of ρ11 is nonexponential because its inversion levelis positive, thus the electric field emitted initially grows inmagnitude as a consequence of the initial condition. Once ourQD becomes an absorber (ρ11 < 0.5), we start to recover theexpected exponential decay shape of radiative decay, whicheventually becomes fully exponential at t0. Finally, in Fig. 7(b),we compare our low-Q cavity to the usual high-Q simulation,and find the semiclassical analog of vacuum Rabi oscillations,namely, we get periodic cycles of the population at a rate givenby 2g = Rabi [83,89], where g is the QD-cavity couplingrate (which scales with d2/Veff) and Rabi is the width of thefrequency splitting in frequency space.

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VI. MAIN SIMULATION RESULTS OF GAIN THRESHOLDFOR A QD ENSEMBLE IN PC CAVITIES

OF DIFFERENT LENGTH

A. QD ensemble

Using the QD ensemble defined above, we are now readyto simulate gain in the 2D PC cavities, in a self-consistent way.The simulation domain is described in Sec. V [see Fig. 5(b)].The only parameter left undefined is the inhomogeneousensemble’s peak frequency ω0. As the homogeneous PC has aband gap between 185–215 THz, the peak frequency should besomewhere within this range. We define two different centerω0 values to study two different gain models: ω1 = 187 THz(gain model A) and ω2 = 197 THz (gain model B), whichresults in two different gain spectra from the QD ensemble,shown in Fig. 8(b). These two values were chosen to representtwo different gain regimes: mainly single-mode lasing overall cavity lengths (which was seen and reported in Ref. [13])performed by ω1 (model A), as the mode nearest resonanceis always the first fundamental mode; and a peak frequencydetermined by the electroluminescence for the QD used inRef. [13], performed by ω2 (model B). Simulations with ω1

and ω2 are carried out with all other parameters equal forconsistency. Figure 8(a) compares the peak frequency of thefirst five fundamental modes as a function of cavity lengthto the resonant frequency of the two QD ensembles fromFig. 8(b).

N

M1M2

M3M4

M5

FIG. 8. (a) Dependence of the cavity mode peak frequencies,for each cavity length, to the resonant peak QD ensemble gain (orPL) spectra. (b) Visualization of the normalized imhomogeneouslybroadened gain spectra of the QD ensemble for ω1 (gain model A)and ω2 (gain model B) taken from Fig. 6(a).

FIG. 9. Lasing dynamics (field in arbitrary units) of an L5 cavitygiven a QD ensemble with ω1 (gain model A), as measured at thecenter of the cavity (x = y = 0). The pump rate of 0.1 ns−1 doesnot lase and only decays, while a pump rate 1 ns−1 is into the lasingregime, as it increases in amplitude, followed by decay as it finds itsequilibrium, and a pump rate 8 ns−1 is clearly well into the lasingregime.

Although the pumped QDs do not need any additionaloptical source to achieve lasing, we find SS can be reachedmuch more quickly when an external (linear) plane-wavesource [seen in Fig. 5(b)] initially excites the cavity. Thedownside of this approach is that some amount of powerwill always be captured by the power monitor, which isdiscussed in the next subsection. However, the lasing thresholdis determined by the slope taken from simulations that areclearly lasing. The dynamics of each simulation is obviouslydifferent depending on if the cavity is below, near, or abovethreshold, as is shown in Fig. 9, where a pump rate of 0.1 ns−1

only decays, while 1 ns−1 increases, followed by decay as itfinds its equilibrium, and 8 ns−1 is well into the lasing regime,finding equilibrium quickly.

The pump threshold Pth is defined by the usual method ofextending the linear region of a “light-in–light-out” (LL) curvedown to the x axis, as shown in Fig. 10 for ω1 (model A) [theresults for ω2 (model B) are qualitatively similar]. Comparingthe low-resolution transmission spectra, measured along a lineat the end of the PC membrane as depicted in Fig. 5(b), inSec. V, for the L7 and L15 cavities of these two different QDensembles, we see that ω1 is predominantly a single-mode laser[Figs. 11(a) and 11(b)], while ω2 is more clearly multimode[Figs. 11(c) and 11(d)]. A major advantage of our model isthat is is able to capture all optical modes that appear in thelight-matter coupling.

B. Role of QD ensemble characteristics and a nonuniformradiative decay rate

The pump thresholds for each cavity length were firstperformed with several QD instances to check for significantfluctuations in any of the trend lines. An example of thiscan be seen in Fig. 12(a), where a reasonable variationwas seen between L13 and L15 pump thresholds. All other

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L5

L7

L9

L11

L13

L15

FIG. 10. Example lasing curve, or LL graphs of the ω1 QDensemble, where the dashed lines represent simulated data, and solidlines are the linear fit used to extract pump threshold. The inset is acloseup of the L5, L9, and L13 curves.

pump thresholds represent a single instance, and any erroror uncertainty in the computations of single instance gainthreshold is determined by the maximum and minimumfitted slopes, and uncertainty in the y intercept. There issome uncertainty in the y intercept due to artificial powermeasurements at low pump strengths, which is caused bythe initial source amplitude continuing to leak into the powermonitor at the end of the simulation, due to the high-Q factorsof the cavities. As such, there exists a positive y intercept forthe measured power emitted by the cavity, when the pump rateis set to zero. This can be seen in Fig. 13, which is an L9

L7

L7

L7

L15

L15

L7 L15

L15

FIG. 11. (a), (b) Example transmission measured at the outputpower DFT monitor for lasing L7 (a) and L15 (b) cavities (P =8 ns−1) the ω1 QD ensemble. (c), (d) Example transmission measuredwith the ω2 QD ensemble.

N

N N

N

FIG. 12. Pump gain thresholds extracted from LL curves forvarious setups. Error bars on individual instances are determined bythe maximum and minimum fitted slopes, and any uncertainty in the y

intercept due to artificial power measurements at low pump strengthscaused by early terminations of the simulations (as demonstrated inFig. 13). (a) The average (dotted black line) of only two QD instances(orange lines) of ω1. (b) A comparison of ω1 (gain model A) and ω2

(gain model B) simulations. (c) A comparison of the sheet simulationto the average QD simulations. (d) Thresholds given by simulationswith a phenomenological radiative decay rate �R , set to 0.05 THz,for all QDs, for both ω1 and ω2 gain models.

simulation from one of the two QD instances for ω1 plotted inFig. 12(a), with an originally negative pump threshold due tothis artifact . To remove the artifact, we fit the low pump datato a polynomial curve, and extract a fitted y intercept, whichwe use to shift the original data to have a y intercept of zero.The error of this fit is then added in quadrature to the slopeserror. The average threshold trend is shown in Fig. 12(a) forω1 simulations and compared to ω2 in Fig. 12(b).

Next, focusing on the dominant single-mode lasing regime,we compare the QD ensemble with simulations that excludethe ensemble statistics by replacing the active gain region witha single plug-in sheet that uniformly excites the various cavitymodes. This “sheet” simulation has a pure dephasing valueequal to the inhomogeneously broadened ensemble of 10 THz,a peak frequency of ω1, and a dipole moment d = 5.84 D(which was found to model the average dipole moment of theensemble), and Neff = 1.045 × 105 μm−3, which was used asa fitting parameter to get the L5 pump threshold behaviorsimilar to those with the QD ensemble value. The resulting

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FIG. 13. Example of y-intercept artifact in our simulations, whichis due to the large-Q factor of our cavities that inherently emit trappedradiation beyond the end of the finite-time simulation.

pump threshold is shown in Fig. 12(c), compared to the averageQD ensemble, which shows a lesser plateauing effect, andincreasing pump threshold for increasing cavity lengths. Weobserve that the spatial-dependent coupling of radiative decayand gain coupling certainly has a qualitatively important roleon the gain threshold characteristics and such behavior wouldbe extremely difficult to capture in a simplified rate-equationanalysis. Moreover, our findings, though somewhat surprising,are indeed consistent with the unusual experimental trendsfound by Xue et al. [13], who attributed the threshold behaviorto structural disorder.

To better understand the effects of the FDTD computedradiative decay rates as a function of cavity length (and thegeneral nonuniform sampling of a radiative decay rate), inFig. 12(d) we have introduced an additional phenomenologicalradiative decay rate into the OBEs, �R using the Lindbladsuperoperator �RL(σ−)ρ, and set �R = 0.05 THz (50 ps−1),which is roughly 100 times smaller than the maximally coupledQD decay rate, averaged over all cavity lengths. However,typically, this will now be the dominant radiative decayprocess in the simulation. By reducing the natural cavitylength dependence of the radiative decay captured by theFDTD method, we see the effects of cavity resonance couplingmore clearly, which is reflected in the pump threshold trendin Fig. 12(d). That is, the trend of model A’s pump thresholdnow consistently decreases, as the on-resonant peak cavity PFincreases and the resonant mode M1 does not significantlychange its frequency position with respect to ω1. On the otherhand, the trend of ω2 is clearly impacted by how close anyparticular mode is to resonance (e.g., highlighted by a dip inthe threshold between L9 and L11, as resonance conditionsare met, and a missing dip between L5 and L7, as resonanceconditions are removed). In either case, this fixed �R , eventhough much smaller than γ ′, clearly has a qualitative influenceon the gain threshold characteristics and thus the nonuniformsampling of such an effect is important.

C. Discussion and connection to simplified laser rate equations

Although it is difficult to identify the main processresponsible for explaining the qualitative difference in gain

threshold behavior shown in Fig. 12(c) (with a plateau) andFig. 12(d) (monotonous reduction), we speculate as follows: If�R and the gain for all QD emitters is forced to be identical (i.e.,not captured self-consistently), all emitters then contribute tothe pump rate and the gain. However, in the full calculationsonly a fraction of the emitters actually contribute to the gainsince such values are still calculated from the actual field fromthe self-consistent FDTD algorithm. Thus, only the fractionof emitters contributing to the gain will be affected when thecavity loss rate is changed with different cavity lengths. Forexample, if the cavity length is increased and the Q factordecreases, less gain is needed for lasing and the pump thresholdwill decrease, but there is still a large background pump ratedue to all the emitters that do not contribute to the gain.Thus, we suggest that if there is a fixed (and dominant) �R ,then this relatively large background pump rate may act tomask small absolute changes in the threshold gain [compareabsolute scales in Figs. 12(c) and 12(d)]. A possible reasonfor the plateauing of the gain threshold is that the effectivegain seen by the QDs changes substantially depending uponthe cavity length. We elaborate on this point in more detailbelow by connecting to common cavity laser rate equations.In Ref. [13], it was shown that the experimental results couldbe explained if one assumes a cavity loss rate that increaseswith the group refractive index. However, this assumption, atleast for the passive cavity, is challenged by the simulationresults of Fig. 1. Using the simple rate equations, e.g., ofPrieto et al. [28] for the carrier and photon densities of a QDPC cavity laser system (neglecting nonradiative decay in thecarrier density equation), then

dNe

dt= Rp − Ne�R − gceff

(Ne − Ne

0

)Nph, (19)

dNph

dt= �cfgceff

(Ne − Ne

0

)Nph + �cfβNe�R − Nph�c,

(20)

where ceff = c/neff, g is the differential gain coefficient, �c =ωc/Q, and �cf is an effective confinement factor. From theseequations, they estimate an approximate pumping thresholdfor lasing:

Pth ∝ Rthp ∝ Ne

th(1 − β)�R, (21)

with the corresponding threshold density

Neth = Ne

0 + �c(N )

�cf(N )ceffg

= N0 + ωc(N )

Q(N )�cf(N )ceffg, (22)

suggesting that Neth goes down as Q goes up (longer cavities),

implying that Pth should come down for increasing cavitylengths. However, one also has to recognize that the �cf(N ),with N the cavity length, will involve a complex spectral (andspatial) coupling between the QDs and cavity mode(s). Oursimulations point to a clear failure of such simple rate-equationapproaches for these complex nanophotonic cavity systems,that is perhaps caused by an overall decrease in the effectivegain caused by a spectral sharpening of the cavity modes,in addition to possible disorder-induced scattering effects. To

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gain some insight into how the LDOS changes at the QD sitesfor each cavity, below we show the main cavity mode radiativedecay rates.

D. Role of spatially varying radiative decay rates

So far, we have exemplified our self-consistent numericalapproach that agrees quite well with experimental gainthreshold behavior without any structural disorder (whichsomewhat contradicts earlier assumptions), yet it is not clearwhy this happens. In an attempt to understand the changingradiative decay rates (and more generally the LDOS, whichwill also affect the QD gain) from an analytical perspective,we can easily use the fundamental QNM spatial profiles tocompute �a(ra), using Eq. (18), and statistically sample awide range of spatial points subjected to the same frequencyaveraging as described above for the QD ensemble (eachone assigned a random center frequency). In Fig. 14, wedemonstrate the M1 projected radiative decay rates sampledover a wide range of spatial points with random frequenciesassigned as before; namely, we use the QNM expansion of theGF, through Eq. (1), using just this normalized cavity mode.

FIG. 14. For each cavity length, we show an example instance ofthe bar-graph statistics from 300 bins of the single-position radiativedecay rates overlapping with the fundamental (M1) cavity mode,where the center frequencies of the dipole emitters are randomlyassigned as before. The total average enhancement rates for eachstatistical distribution, using all spatial points in the spatial modeactive region, is shown in each panel (in units of γ0).

N

FIG. 15. Total ensemble average of the single QD radiative decayrate for each LN cavity averaged over 50 statistical variations ofeach instance shown in Fig. 14. In contrast to the peak Purcell factortrends, we clearly see a decrease in the average radiative decay ratefor increasing cavity lengths that would be randomly sampled bythe QDs.

Instead of an increasing decay rate, we see a decrease of theradiative decay rates for increasing length cavities since theprobability of spectrally overlapping with the larger-Q cavitymodes decreases. To help quantify this effect, we have repeatedthese statistical simulations 50 times each and computed theensemble spatial average per LN cavity and confirm thatthere is indeed a decrease as shown in Fig. 15. Thus, itseems likely that this decreasing radiation decay rate causesan increase in the threshold gain characteristics, and perhapswill plateau as the average increase falls below γ0 (associatedwith out-of-plane decay of the PC slab). While the rates aremuch smaller than γ ′ (1.5 ps−1), they still play a key role indetermining the overall population decay. This view is furthersupported by the calculations shown in Fig. 12(d), wherea fixed (and dominant) radiative decay rate �R = 0.05 ps−1

(�γ ′) increases the threshold currents by at least an orderof magnitude [cf. Fig. 12(c)], and completely changes thethreshold gain characteristics as a function of cavity length.

VII. CONCLUSIONS

We have developed a self-consistent numerical model todescribe an active QD ensemble coupled to PC cavities usingLumerical’s plug-in tool within an effective 2D FDTD methodto investigate pump threshold as a function of cavity length.Both multimode and (primarily) single-mode lasing wasfound, depending on the peak frequency of the QD ensemble.Studying the effects of our ensemble on the single-mode lasing,we found a general plateauing (and possible increase) of thepump threshold beyond L9, in qualitative agreement withrecent experiments. As such, we believe there is strong desireto generalize the common rate equations for these complexcavity systems. From a simple analytical modal theory of thefundamental cavity mode, we have also demonstrated how theoverall radiative decay rates (and likely other related effectssuch as the effective gain) can come down as a function ofcavity size, which is caused by a reduction in the spectraloverlap of the spectrally sharp cavity modes with respectto the broad emission frequencies of the QD ensemble. Inthe Appendix, structural disorder is also shown to generallyincrease the pump threshold for cavities longer than L7,again in good agreement with the experimental findings of

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Ref. [13], though it seems unlikely that disorder alone wasresponsible for this effect (as our simulations without disorderclearly show). Random localization due to disorder of thelasing cavities modes is also seen, which could merit furtherinvestigation, e.g., in the context of slow-light PC waveguides[58,90] and active waveguides [91].

ACKNOWLEDGMENTS

This work was supported by the Natural Sciences andEngineering Research Council of Canada and Queen’sUniversity. We also gratefully acknowledge the help ofLumerical’s support team, as well as E. Schelew and N.Mann. This research was enabled in part by computationalsupport provided by WestGrid (www.westgrid.ca), ComputeCanada (www.computecanada.ca), and Lumerical Solutions(www.lumerical.com). J. Mørk acknowledges support fromVillum Fonden via the NATEC Center of Excellence (GrantNo. 8692).

APPENDIX: EXAMPLES OF SINGLE-INSTANCEDISORDERED LATTICES ON THE GAIN

THRESHOLD BEHAVIOR

In this Appendix, we briefly assess the impact of single-instance structural disorder on the lasing pump threshold,namely, in the presence of the self-consistent gain dynamics.As mentioned in Sec. II, the intrinsic level of disorder forour effective 2D simulations is given by σDis = 0.01a (whichmimic the same effect as full 3D slab cavities properties).To understand the effects of additional lattice disorder, weincrease this to σDis = 0.04a. Instances of these two disorderedsimulations can be seen in Figs. 16(a) and 16(b), wherewe again use ω1 as the peak frequency (gain model A).A general increase to the pump threshold, which becomesmore prominent beyond L7, is clearly seen. These resultsare indeed consistent with the experimental results from [13],showing that the laser threshold plateaus for increasing cavity

N N

FIG. 16. Pump threshold trends for two disordered simulations:intrinsic levels σDis = 0.01a (a), and larger than intrinsic disorderσDis = 0.04a (b), demonstrating a general increase in the pumpthreshold predominantly for cavity lengths greater than L7.

length. Our results show that the inhomogeneous nature of theensemble of quantum dots plays a strong role in this effect,and may even dominate the effects of structural disorder.

Finally, we also look at the influence of increased disorder(e.g., deliberate disorder) on our lasing mode profiles in thepossible regime of Anderson localization, or with stronglylocalized modes. At L15, the M1 is the strongest mode

FIG. 17. (a) Projected LDOS along the y direction for the firstfundamental lasing mode of the L15 cavity, for the sheet active regionwith peak frequency ω1, with increasing amounts of disorder. TheLDOS is normalized to free space such that our y axis is in units ofPF. All lasing modes are measured for simulations with incoherentpump rate P = 8 ns−1, where the ordered simulation’s response isin black, the intrinsically disordered simulation σDis = 0.01a is inorange, and the additional disordered simulation σDis = 0.04a is inmagenta. (b)–(d) Mode profiles of M1 for the ordered (b), intrinsicdisordered σDis = 0.01a (c), and additional disordered σDis = 0.04a

(d) simulations.

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(highest Q) of all the cavity lengths, and also has thelongest mode profile, which makes it the best candidate forlocalization. Sheet (uniform) gain is used in this analysis toisolate the randomness in the systems to the disorder of the PC.Figures 17(b)–17(d) depict the localization of the fundamentalmode as disorder is increased, although the overall PF is stillless than the idealized structure as shown in Fig. 17(a). Theprojected LDOS is measured at the peak antinode location

for each mode, which is marked in each mode profile by thesmall black “x” marker. This localization is random in nature,which potentially limits the applications of such a mode, andalthough the mode volume is reduced, the Q factor takes aneven greater hit. Note with greater disorder comes a larger Q

variance, as seen in Fig. 4(d), which does mean it is possiblefor these localized modes to have reduced mode volume andincreased Q factors.

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